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Fair division of indivisible bads seeks allocations that guarantee every agent a bundle whose cost is no larger than a meaningful fairness benchmark. The canonical minimax share has widely been used; unfortunately, it is not a simultaneous guarantee. Hill (Ann. Probab., 1987) initiated a complementary approach in which...
Friedgut conjectured that an increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have bounded size, with a bound independent of the dimension and the bias (J. Amer. Math. Soc. 12 (1999)). We prove this conjecture by showing...
If each coordinate appears in only a small number of clauses, there is a quantum algorithm for strongly log-concave sampling using local queries using $\widetilde{O}(\sqrt{\kappa}d)$ local queries, where $\kappa$ is the condition number.
It is proved that Block Orthogonality alone forces both Type Partition and the existence of a single Mal'tsev operation preserving all generated support and row-equivalence relations, and the three-condition characterization collapses to Block Orthogonality.
We prove that counting perfect matchings is $\#P$-complete under polynomial-time Turing reductions on each of three classes of simple, unweighted graphs: monotone graphs, unit interval graphs, and chordal permutation graphs. The monotone result settles the exact-counting complexity left open by Dyer, Jerrum, and M\"ull...
The rank-independent theorem sharpens many later guarantees that inherit their sampling bounds by strengthening the independent STOC 2023 works of Lee and Jambulapati--Liu--Sidford by removing their rank dependence and answering Lee's open question on whether this loss is inherent.
A dimension-free version of the retained-energy form of the Mallat--Zeitouni conjecture is established, showing that the KL basis is within this factor of the optimal basis, and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows.
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